Triangle 
\[\triangle A'B'C'\] is the image of 
\[\triangle ABC\] under a rotation about the origin, 
\[(0,0)\].

\[\small{1}\]
\[\small{2}\]
\[\small{3}\]
\[\small{4}\]
\[\small{5}\]
\[\small{6}\]
\[\small{7}\]
\[\small{\llap{-}2}\]
\[\small{\llap{-}3}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}5}\]
\[\small{\llap{-}6}\]
\[\small{\llap{-}7}\]
\[\small{1}\]
\[\small{2}\]
\[\small{3}\]
\[\small{4}\]
\[\small{5}\]
\[\small{6}\]
\[\small{7}\]
\[\small{\llap{-}2}\]
\[\small{\llap{-}3}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}5}\]
\[\small{\llap{-}6}\]
\[\small{\llap{-}7}\]
\[y\]
\[x\]
\[A\]
\[B\]
\[C\]
\[A'\]
\[B'\]
\[C'\]
Determine the angle of rotation.
Choose 1 answer:
Choose 1 answer:
(Choice A)   
\[-60^\circ\]
A
\[-60^\circ\]
(Choice B)   
\[-30^\circ\]
B
\[-30^\circ\]
(Choice C)   
\[30^\circ\]
C
\[30^\circ\]
(Choice D)   
\[60^\circ\]
D
\[60^\circ\]

To determine the angle of rotation of \(\triangle ABC\) to \(\triangle A'B'C'\) about the origin, we need to consider how the coordinates of the points are transformed. 

Let the coordinates of points \(A\), \(B\), and \(C\) in the original triangle be \((x, y)\). After rotation, the new coordinates of the points, \((x', y')\), can be determined using the rotation matrix:

\[
\begin{pmatrix}
x' \\
y'
\end{pmatrix}
=
\begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]

where \(\theta\) is the angle of rotation. The signs and values of \(x'\) and \(y'\) will help identify the rotation angle.

The choices provided suggest possible angles, and we need to determine which one matches the transformation from the original triangle to its image. The positive or negative sign of the angle indicates the direction of rotation: positive for counterclockwise and negative for clockwise.

Without the specific coordinates, we can't determine the exact angle just by the general transformation formula. However, the provided information about choosing one of the angles gives a hint about the direction and magnitude of rotation.

### Answer:
To solve this problem accurately, I need to see the coordinates of the points in both triangles before and after rotation. If you can provide the coordinates of any corresponding points (like \(A\) and \(A'\), \(B\) and \(B'\), or \(C\) and \(C'\)), I can use them to determine the exact angle of rotation.

Alternatively, if you describe the changes in coordinates or their positions relative to the origin, that information will help as well.

Do you want more details or have any questions?

### Relative Questions:
1. How are rotations represented in a coordinate plane?
2. What is the significance of the origin in rotations?
3. How do you determine the direction of rotation (clockwise vs. counterclockwise)?
4. What is the role of the rotation matrix in transforming coordinates?
5. How can you identify the angle of rotation if given the coordinates before and after rotation?
6. What are the possible rotation angles that result in a triangle mapping to its image?
7. What is the effect of a negative angle in rotation transformations?
8. How can geometric properties like lengths and angles help confirm the rotation angle?

### Tip:
When working with rotations, always pay attention to the direction (clockwise or counterclockwise) as it affects the sign of the angle in the transformation.

Explore how to determine the angle of rotation between two triangles, \(\triangle ABC\) and its image \(\triangle A'B'C'\), under rotation about the origin. Understand the transformation using rotation matrices and how to identify the correct angle of rotation. Suitable for high school students studying geometry and coordinate transformations.

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